CHAPTER 10: TIME-VARYING FIELDS AND MAXWELL''''S EQUATIONS pot

In terms of fields, we now say that a time-varying magnetic field produces an electromotive force emf which may establish a current in a suitable closed circuit.. The magnetic flux is th

CHAPTER 10 TIME-VARYING FIELDS AND MAXWELL'S EQUATIONS

The basic relationships of the electrostatic and the steady magnetic field wereobtained in the previous nine chapters, and we are now ready to discuss time-varying fields The discussion will be short, for vector analysis and vector calcu-lus should now be more familiar tools; some of the relationships are unchanged,and most of the relationships are changed only slightly

Two new concepts will be introduced: the electric field produced by achanging magnetic field and the magnetic field produced by a changing electricfield The first of these concepts resulted from experimental research by MichaelFaraday, and the second from the theoretical efforts of James Clerk Maxwell.Maxwell actually was inspired by Faraday's experimental work and by themental picture provided through the ``lines of force'' that Faraday introduced indeveloping his theory of electricity and magnetism He was 40 years youngerthan Faraday, but they knew each other during the 5 years Maxwell spent inLondon as a young professor, a few years after Faraday had retired Maxwell'stheory was developed subsequent to his holding this university position, while hewas working alone at his home in Scotland It occupied him for 5 years betweenthe ages of 35 and 40

The four basic equations of electromagnetic theory presented in this ter bear his name

chap-322

10.1 FARADAY'S LAW

After Oersted1demonstrated in 1820 that an electric current affected a compass

needle, Faraday professed his belief that if a current could produce a magnetic

field, then a magnetic field should be able to produce a current The concept of

the ``field'' was not available at that time, and Faraday's goal was to show that a

current could be produced by ``magnetism.''

He worked on this problem intermittently over a period of ten years, until

he was finally successful in 1831.2He wound two separate windings on an iron

toroid and placed a galvanometer in one circuit and a battery in the other Upon

closing the battery circuit, he noted a momentary deflection of the galvanometer;

a similar deflection in the opposite direction occurred when the battery was

disconnected This, of course, was the first experiment he made involving a

changing magnetic field, and he followed it with a demonstration that either a

moving magnetic field or a moving coil could also produce a galvanometer

deflection

In terms of fields, we now say that a time-varying magnetic field produces

an electromotive force (emf) which may establish a current in a suitable closed

circuit An electromotive force is merely a voltage that arises from conductors

moving in a magnetic field or from changing magnetic fields, and we shall define

it below Faraday's law is customarily stated as

Equation (1) implies a closed path, although not necessarily a closed conducting

path; the closed path, for example, might include a capacitor, or it might be a

purely imaginary line in space The magnetic flux is that flux which passes

through any and every surface whose perimeter is the closed path, and d=dt

is the time rate of change of this flux

A nonzero value of d=dt may result from any of the following situations:

1 A time-changing flux linking a stationary closed path

2 Relative motion between a steady flux and a closed path

3 A combination of the two

The minus sign is an indication that the emf is in such a direction as to

produce a current whose flux, if added to the original flux, would reduce the

magnitude of the emf This statement that the induced voltage acts to produce an

opposing flux is known as Lenz's law.3

If the closed path is that taken by an N-turn filamentary conductor, it isoften sufficiently accurate to consider the turns as coincident and let

emf ˆ

I

and note that it is the voltage about a specific closed path If any part of the path

is changed, generally the emf changes The departure from static results is clearlyshown by (3), for an electric field intensity resulting from a static charge dis-tribution must lead to zero potential difference about a closed path In electro-statics, the line integral leads to a potential difference; with time-varying fields,the result is an emf or a voltage

Replacing  in (1) by the surface integral of B, we have

be kept in mind during flux integrations and emf determinations

Let us divide our investigation into two parts by first finding the tion to the total emf made by a changing field within a stationary path (trans-former emf), and then we will consider a moving path within a constant(motional, or generator, emf)

contribu-We first consider a stationary path The magnetic flux is the only varying quantity on the right side of (4), and a partial derivative may be takenunder the integral sign,

time-emf ˆ

I

E
dL ˆ

ZS

@B

Before we apply this simple result to an example, let us obtain the point

form of this integral equation Applying Stokes' theorem to the closed line

integral, we have

Z

S…r  E†
dS ˆ

ZS

@B

@t
dSwhere the surface integrals may be taken over identical surfaces The surfaces are

perfectly general and may be chosen as differentials,

…r  E†
dS ˆ @B

@t
dSand

r  E ˆ @B

This is one of Maxwell's four equations as written in differential, or point,

form, the form in which they are most generally used Equation (5) is the integral

form of this equation and is equivalent to Faraday's law as applied to a fixed

path If B is not a function of time, (5) and (6) evidently reduce to the

electro-static equations,

I

E
dL ˆ 0 (electrostatics)and

r  E ˆ 0 (electrostatics)

As an example of the interpretation of (5) and (6), let us assume a simple

magnetic field which increases exponentially with time within the cylindrical

region  < b,

where B0 ˆ constant Choosing the circular path  ˆ a, a < b, in the z ˆ 0 plane,

along which E must be constant by symmetry, we then have from (5)

emf ˆ 2aEˆ kB0ekta2The emf around this closed path is kB0ekta2 It is proportional to a2, because

the magnetic flux density is uniform and the flux passing through the surface at

any instant is proportional to the area

If we now replace a by ,  < b, the electric field intensity at any point is

Let us now attempt to obtain the same answer from (6), which becomes

2kB0ektaonce again

If B0 is considered positive, a filamentary conductor of resistance R wouldhave a current flowing in the negative a direction, and this current wouldestablish a flux within the circular loop in the negative az direction Since Eincreases exponentially with time, the current and flux do also, and thus tend toreduce the time rate of increase of the applied flux and the resultant emf inaccordance with Lenz's law

Before leaving this example, it is well to point out that the given field B doesnot satisfy all of Maxwell's equations Such fields are often assumed (always inac-circuit problems) and cause no difficulty when they are interpreted properly.They occasionally cause surprise, however This particular field is discussedfurther in Prob 19 at the end of the chapter

Now let us consider the case of a time-constant flux and a moving closedpath Before we derive any special results from Faraday's law (1), let us use thebasic law to analyze the specific problem outlined in Fig 10.1 The closed circuitconsists of two parallel conductors which are connected at one end by a high-resistance voltmeter of negligible dimensions and at the other end by a sliding barmoving at a velocity v The magnetic flux density B is constant (in space andtime) and is normal to the plane containing the closed path

ly small high-resistance voltmeter.

Bvd.

Let the position of the shorting bar be given by y; the flux passing through

the surface within the closed path at any time t is then

 ˆ BydFrom (1), we obtain

dy

The emf is defined asHE
dL and we have a conducting path; so we may

actually determine E at every point along the closed path We found in

electro-statics that the tangential component of E is zero at the surface of a conductor,

and we shall show in Sec 10.4 that the tangential component is zero at the

surface of a perfect conductor … ˆ 1† for all time-varying conditions This is

equivalent to saying that a perfect conductor is a ``short circuit.'' The entire

closed path in Figure 10.1 may be considered as a perfect conductor, with the

exception of the voltmeter The actual computation ofHE
dL then must involve

no contribution along the entire moving bar, both rails, and the voltmeter leads

Since we are integrating in a counterclockwise direction (keeping the interior of

the positive side of the surface on our left as usual), the contribution E
L across

the voltmeter must be Bvd, showing that the electric field intensity in the

instrument is directed from terminal 2 to terminal 1 For an up-scale reading,

the positive terminal of the voltmeter should therefore be terminal 2

The direction of the resultant small current flow may be confirmed by

noting that the enclosed flux is reduced by a clockwise current in accordance

with Lenz's law The voltmeter terminal 2 is again seen to be the positive

ter-minal

Let us now consider this example using the concept of motional emf The

force on a charge Q moving at a velocity v in a magnetic field B is

F ˆ Qv  B

or

F

The sliding conducting bar is composed of positive and negative charges, and

each experiences this force The force per unit charge, as given by (10), is called

the motional electric field intensity Em,

If the moving conductor were lifted off the rails, this electric field intensity would

force electrons to one end of the bar (the far end) until the static field due to these

charges just balanced the field induced by the motion of the bar The resultanttangential electric field intensity would then be zero along the length of the bar.The motional emf produced by the moving conductor is then

I

…v  B†
dL ˆ

Z 0

d vB dx ˆ Bvd

as before This is the total emf, since B is not a function of time

In the case of a conductor moving in a uniform constant magnetic field, wemay therefore ascribe a motional electric field intensity Emˆ v  B to everyportion of the moving conductor and evaluate the resultant emf by

and either can be used to determine these induced voltages

Although (1) appears simple, there are a few contrived examples in whichits proper application is quite difficult These usually involve sliding contacts orswitches; they always involve the substitution of one part of a circuit by a newpart.4As an example, consider the simple circuit of Fig 10.2, containing severalperfectly conducting wires, an ideal voltmeter, a uniform constant field B, and aswitch When the switch is opened, there is obviously more flux enclosed in thevoltmeter circuit; however, it continues to read zero The change in flux has notbeen produced by either a time-changing B [first term of (14)] or a conductormoving through a magnetic field [second part of (14)] Instead, a new circuit hasbeen substituted for the old Thus it is necessary to use care in evaluating thechange in flux linkages

The separation of the emf into the two parts indicated by (14), one due tothe time rate of change of B and the other to the motion of the circuit, is some-

what arbitrary in that it depends on the relative velocity of the observer and the

system A field that is changing with both time and space may look constant to

an observer moving with the field This line of reasoning is developed more fully

in applying the special theory of relativity to electromagnetic theory.5

\ D10.1 Within a certain region,  ˆ 10 11 F=m and  ˆ 10 5 H=m If B x ˆ

2  10 4 cos 10 5 t sin 10 3 y T: (a) use r  H ˆ @E@t to find E; (b) find the total magnetic

flux passing through the surface x ˆ 0, 0 < y < 40 m, 0 < z < 2 m, at t ˆ 1 s; (c) find

the value of the closed line integral of E around the perimeter of the given surface.

Ans 20 000 sin 10 5 t cos 10 3 ya z V=m; 31:4 mWb; 315 V

\ D10.2 With reference to the sliding bar shown in Figure 10.1, let d ˆ 7 cm,

B ˆ 0:3a z T, and v ˆ 0:1a y e 20y m=s Let y ˆ 0 at t ˆ 0 Find: (a) v…t ˆ 0†; (b)

which shows us that a time-changing magnetic field produces an electric field

Remembering the definition of curl, we see that this electric field has the special

property of circulation; its line integral about a general closed path is not zero

Now let us turn our attention to the time-changing electric field

We should first look at the point form of AmpeÁre's circuital law as it

applies to steady magnetic fields,

FIGURE 10.2

An apparent increase in flux linkages does not lead to an induced voltage when one part of a circuit is simply sub- stituted for another by opening the switch No indication will be observed

on the voltmeter.

See Panofsky and Phillips, pp 142±151; Owen, pp 231±245; and Harman in several places.

r  H ˆ J …16†and show its inadequacy for time-varying conditions by taking the divergence ofeach side,

r
r  H  0 ˆ r
JThe divergence of the curl is identically zero, so r
J is also zero However, theequation of continuity,

r
J ˆ @v

@tthen shows us that (16) can be true only if @v=@t ˆ 0 This is an unrealisticlimitation, and (16) must be amended before we can accept it for time-varyingfields Suppose we add an unknown term G to (16),

r  H ˆ J ‡ GAgain taking the divergence, we have

0 ˆ r
J ‡ r
GThus

r
G ˆ@v

@tReplacing v by r
D,

r
G ˆ @

@t…r
D† ˆ r

@D

@tfrom which we obtain the simplest solution for G,

G ˆ@D

@tAmpeÁre's circuital law in point form therefore becomes

r  H ˆ J ‡@D

Equation (17) has not been derived It is merely a form we have obtainedwhich does not disagree with the continuity equation It is also consistent with allour other results, and we accept it as we did each experimental law and theequations derived from it We are building a theory, and we have every right

to our equations until they are proved wrong This has not yet been done

We now have a second one of Maxwell's equations and shall investigate itssignificance The additional term @D=@t has the dimensions of current density,amperes per square meter Since it results from a time-varying electric flux den-sity (or displacement density), Maxwell termed it a displacement current density

We sometimes denote it by Jd:

r  H ˆ J ‡ Jd

Jd ˆ@D

@tThis is the third type of current density we have met Conduction current density,

J ˆ E

is the motion of charge (usually electrons) in a region of zero net charge density,

and convection current density,

J ˆ vv

is the motion of volume charge density Both are represented by J in (17) Bound

current density is, of course, included in H In a nonconducting medium in which

no volume charge density is present, J ˆ 0, and then

Again the analogy between the intensity vectors E and H and the flux

density vectors D and B is apparent Too much faith cannot be placed in this

analogy, however, for it fails when we investigate forces on particles The force

on a charge is related to E and to B, and some good arguments may be presented

showing an analogy between E and B and between D and H We shall omit them,

however, and merely say that the concept of displacement current was probably

suggested to Maxwell by the symmetry first mentioned above.6

The total displacement current crossing any given surface is expressed by

the surface integral,

Id ˆ

Z

SJd
dS ˆ

ZS

@D

@t
dSand we may obtain the time-varying version of AmpeÁre's circuital law by inte-

grating (17) over the surface S,

@D

@t
dS

Suggested References for Chap 5) on pp 159±160 and 179; the case for comparing B to E and D to H

is presented in Halliday and Resnick (see Suggested References for this chapter) on pp 665±668 and 832±

836.

and applying Stokes' theorem,

I

H
dL ˆ I ‡ Id ˆ I ‡

ZS

@D

What is the nature of displacement current density? Let us study the simplecircuit of Fig 10.3, containing a filamentary loop and a parallel-plate capacitor.Within the loop a magnetic field varying sinusoidally with time is applied toproduce an emf about the closed path (the filament plus the dashed portionbetween the capacitor plates) which we shall take as

emf ˆ V0cos !tUsing elementary circuit theory and assuming the loop has negligible resis-tance and inductance, we may obtain the current in the loop as

I ˆ !CV0sin !t

ˆ !S

d V0sin !twhere the quantities , S, and d pertain to the capacitor Let us apply AmpeÁre'scircuital law about the smaller closed circular path k and neglect displacementcurrent for the moment:

I

kH
dL ˆ Ik

FIGURE 10.3

A filamentary conductor forms a loop connecting the two plates of a parallel-plate capacitor A

conduction current I is equal to the displacement current between the capacitor plates.

The path and the value of H along the path are both definite quantities (although

difficult to determine), andHkH
dL is a definite quantity The current Ikis that

current through every surface whose perimeter is the path k If we choose a

simple surface punctured by the filament, such as the plane circular surface

defined by the circular path k, the current is evidently the conduction current

Suppose now we consider the closed path k as the mouth of a paper bag whose

bottom passes between the capacitor plates The bag is not pierced by the

fila-ment, and the conductor current is zero Now we need to consider displacement

current, for within the capacitor

loop Therefore the applicaton of AmpeÁre's circuital law including displacement

current to the path k leads to a definite value for the line integral of H This value

must be equal to the total current crossing the chosen surface For some surfaces

the current is almost entirely conduction current, but for those surfaces passing

between the capacitor plates, the conduction current is zero, and it is the

dis-placement current which is now equal to the closed line integral of H

Physically, we should note that a capacitor stores charge and that the

electric field between the capacitor plates is much greater than the small leakage

fields outside We therefore introduce little error when we neglect displacement

current on all those surfaces which do not pass between the plates

Displacement current is associated with time-varying electric fields and

therefore exists in all imperfect conductors carrying a time-varying conduction

current The last part of the drill problem below indicates the reason why this

additional current was never discovered experimentally This comparison is

illu-strated further in Sec 11.3

\ D10.3 Find the amplitude of the displacement current density: (a) adjacent to an

automobile antenna where the magnetic field intensity of an FM signal is H x ˆ

0:15 cos‰3:12…3  10 8 t y†Š A=m; (b) in the air space at a point within a large power

distribution transformer where B ˆ 0:8 cos‰1:257  10 6 …3  10 8 t x†Ša y T; (c) within a

large oil-filled power capacitor where  R ˆ 5 and E ˆ 0:9 cos‰1:257  10 6 …3  10 8

t zp5 †Ša x MV=m; (d) in a metallic conductor at 60 Hz, if  ˆ  0 ,  ˆ  0 ,  ˆ

5:8  10 7 S=m, and J ˆ sin…377t 117:1z†a x MA=m 2

Ans 0.318 A/m 2 ; 0.800 A/m 2 ; 0.01502 A/m 2 ; 57.6 pA/m 2

10.3 MAXWELL'S EQUATIONS IN POINT

E, and hence D, may have circulation if a changing magnetic field is present.Thus the lines of electric flux may form closed loops However, the converse isstill true, and every coulomb of charge must have one coulomb of electric fluxdiverging from it

Equation (23) again acknowledges the fact that ``magnetic charges,'' orpoles, are not known to exist Magnetic flux is always found in closed loopsand never diverges from a point source

These for equations form the basis of all electromagnetic theory They arepartial differential equations and relate the electric and magnetic fields to eachother and to their sources, charge and current density The auxiliary equationsrelating D and E

relating B and H,